how can you extract the astaxanthin from the giant tiger prawn?

Does astaxanthin prevent tanning?

Astaxanthin is a powerful antioxidant. There is scant research on astaxanthin in human. There is research on its photoprotecton against UV-rays. However, this is true for many other antioxidants including lycopene, vitamin C, vitamin E, green tea, etc.
Bottom line: your best option is to use a physical sunblock with zinc oxide or titanium dioxide.

Is Astaxanthin good for you?

http://en.wikipedia.org/wiki/Astaxanthin
It is a carotenoid that gives vegetables and crustaceans a red orange coloring. It is used as a food additive and food coloring agent. There are natural and synthetic versions available. It supposedly has lots of antioxidant properties, but the jury is out on whether there is any real clinical benefit from taking antioxidants.
If so, you would likely do better with antioxidants from other sources, like fresh fruits and vegetable rather than as a supplement.

What property/ies does astaxanthin have that enables it pose serious environmental threats?

astaxanthin is a caratenoid, a group of chemicals that is a precursor to vitamin A. It gives colour to salmon, shrimp and flamingos, and is used as a colourant in food for human consumption.Its long chain of double C bindings, as well as the O and OH groups, make that it is reatively easily degraded. At the levels in which it occurs in algea and shrimp, it is non-toxic, and it appears to have beneficial effects to humans as a an antioxidant.
SO to answer your question: i think astaxathin is environmentally benign, and does not pose any environmental threats.

Astaxanthin, is it like vitamins for eyes?

Well astaxanthin is a carotenoid, basically it is a powerful antioxidant. I've provided a link to wikepedia, most of the time people just read the paragraphs, but the bottom section is full of resources related to astaxanthin. I've also provided a link for Google News for that subject.

why does astaxanthin protect you from sunburns? and why do we sunburn?

First, lets put down some information.
In each skin cell, we have what we call melanocytes, which depending on your skin type, develop melanin. Melanin is a pigment that gives our skin color. when exposed to the sun. Everyone is born with the same number of melanocytes, but the darker your skin is, the more light you can withstand without having to boost your melanin production.
When your body senses that you need more melanin to protect you against harmful UV rays, your melanocytes kick into high gear and you get a suntan. However, if you stay outside for too long, especially without sunscreen, then your body can't make melanin fast enough to keep up with the amount of UV exposure. This is what causes a sunburn.
Astaxanthin is a natural pigment found in algae that protects them from sunlight. BioAstin, "the natural Astaxanthin", does and will not prevent sunburning. However, studies have shown that it will increase your body's tolerance of UV exposure by 20-50%, allowing you to be outside longer without developing sunburn.
As anti-sunburn method, sunscreen is your best bet.
.

What are the harmful effects of astaxanthin to the environment?

Usually on an industrial Large scale ,anything can be made harmful to creature in the environment . Imagine dumping a million tons of a carotenoid like astaxanthin into a river .
What do you think would happen to the fish ? Even though carotenoid are a naturally occurring vitamin A, too much of a good think can be a Bad thing . Then , on a very small scale ,using it as a food coloring or dye in small amount can be OK . I hope this helps you .

Astaxanthin, how does it effect other carotenoids ?

Good question! I couldn't think of a good reason why it would impair absorption of other carotenoids, so it sent me to the literature to check. :)
Here's a quote:
"In the plasma, oil-soluble carotenoids such as beta- and alpha-carotene or lycopene, are mostly transported by very low-density lipoproteins (VLDLs) and low-density lipoproteins (LDLs), while more polar carotenoids, such as astaxanthin, zeaxanthin, or lutein, are more likely to be transported by LDLs and high-density lipoproteins (HDLs)."
(http://www.algatech.com/bio.htm)
Also:
http://www.cyanotech.com/pdfs/bioastin/batl13.pdf (which is specific to astaxanthin absorption, but fleshes out the above.)
There's sufficient data out there to support supplementing with astaxanthin as part of general health promotion. My suggestion would be that if you're SUPPLEMENTING with astaxanthin (because really how many of us can eat that much lobster?) then I'd get the rest of my carotenoids from dietary sources. Best absorption that way, anyway! AND you're getting the full compliment of other nutrition in the fruits/veg. AND you're avoiding some of the risks of supplementing, such as increased risk of lung cancer for smokers supplementing beta-carotene.
Incorporate loads of brightly-colored orange, red and yellow fruits and veg. Eat them fresh. Stew them (increases bioavailability of lycopene, for example.) But get them in your diet instead of in a pill.
----------------------------------
Jack - nope. Shouldnt impact absorption at all. Best if you take 'em both with a meal, though. And as I've said above and many times before... I'm not a fan of the multi. I'd rather see you getting your nutrition from your food. If that's not happening, my next best option is a food-based supplement (like Greens + or Berries +, or any of the spirulina-based powders, etc.) Again, when you're taking as close to possible as the whole food, you're getting a broader range of nutrition.
If you must take a multi, find one with as few additives and fillers as possible. We're starting to learn that those fillers can have a negative impact on health.